Search for the $X(2370)$ and observation of ${\eta }_c\to \eta \eta {\eta }^{\mathrm{\prime }}$ in $J\mathrm{/}\psi \to \gamma \eta \eta {\eta }^{\mathrm{\prime }}$

Using a sample of $1.31\times {10}^{9}J\mathrm{/}\psi$ events collected with the BESIII detector, we perform a study of $J\mathrm{/}\psi \to \gamma \eta \eta {\eta }^{\mathrm{\prime }}$ to search for the $X(2370)$ and ${\eta }_c$ in the $\eta \eta {\eta }^{\mathrm{\prime }}$ invariant mass distribution. No significant signal for the $X(2370)$ is observed, and we set an upper limit for the product branching fraction of ${\cal B}(J/\psi\to\gamma X(2370)\cdot{\cal B}(X(2370)\to\eta\eta\eta^{\prime}) < 9.2\times10^{-6}$atthe90$\eta_{c}$ signal is observed for the first time, yielding a product branching fraction of ${\cal B}(J/\psi\to \gamma\eta_{c})\cdot{\cal B}(\eta_{c}\to \eta\eta\eta^{\prime}) = (4.86\pm0.62~({\rm stat.})\pm0.45~({\rm sys.}))\times10^{-5}$.